Corollary

Tag 05N0, permanent. Cite it as [Bourbaki, Tag 05N0].

Algebra, chapter II, § 2. Modules of linear mappings. Duality

For an A-linear mapping $u : E \to F$ to be injective, it is necessary and sufficient that, for every A-module G, the mapping $\operatorname{Hom}(1_G, u) : \operatorname{Hom}(G, E) \to \operatorname{Hom}(G, F)$ be injective.

It suffices to apply Theorem 2 to the case where $F' = \{0\}$ .

Starting from an exact sequence

0 \longrightarrow F' \xrightarrow{u} F \xrightarrow{v} F'' \longrightarrow 0

the corresponding sequence

0 \longrightarrow \operatorname{Hom}(E, F') \xrightarrow{\bar{u}} \operatorname{Hom}(E, F) \xrightarrow{\bar{v}} \operatorname{Hom}(E, F'') \longrightarrow 0

is not necessarily exact, in other words $\bar{v}$ is not necessarily surjective. If $F'$ is identified with a submodule of $F$ and $F''$ with the quotient module $F/F'$ , this means that a linear mapping of $E$ into $F''$ is not necessarily of the form $v \circ w$ , where $w$ is a linear mapping of $E$ into $F$ . However:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Theorem 2